| L(s) = 1 | + 2.69·2-s + 5.24·4-s + 2.40·5-s + 2.96·7-s + 8.72·8-s + 6.47·10-s − 0.991·11-s + 3.08·13-s + 7.97·14-s + 12.9·16-s − 3.39·17-s − 3.05·19-s + 12.6·20-s − 2.66·22-s − 5.70·23-s + 0.791·25-s + 8.31·26-s + 15.5·28-s − 3.78·29-s + 4.30·31-s + 17.5·32-s − 9.13·34-s + 7.13·35-s + 5.83·37-s − 8.20·38-s + 20.9·40-s − 0.306·41-s + ⋯ |
| L(s) = 1 | + 1.90·2-s + 2.62·4-s + 1.07·5-s + 1.12·7-s + 3.08·8-s + 2.04·10-s − 0.299·11-s + 0.856·13-s + 2.13·14-s + 3.24·16-s − 0.823·17-s − 0.699·19-s + 2.82·20-s − 0.569·22-s − 1.19·23-s + 0.158·25-s + 1.63·26-s + 2.93·28-s − 0.702·29-s + 0.773·31-s + 3.09·32-s − 1.56·34-s + 1.20·35-s + 0.959·37-s − 1.33·38-s + 3.31·40-s − 0.0479·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6021 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6021 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.04543060\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.04543060\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 223 | \( 1 - T \) |
| good | 2 | \( 1 - 2.69T + 2T^{2} \) |
| 5 | \( 1 - 2.40T + 5T^{2} \) |
| 7 | \( 1 - 2.96T + 7T^{2} \) |
| 11 | \( 1 + 0.991T + 11T^{2} \) |
| 13 | \( 1 - 3.08T + 13T^{2} \) |
| 17 | \( 1 + 3.39T + 17T^{2} \) |
| 19 | \( 1 + 3.05T + 19T^{2} \) |
| 23 | \( 1 + 5.70T + 23T^{2} \) |
| 29 | \( 1 + 3.78T + 29T^{2} \) |
| 31 | \( 1 - 4.30T + 31T^{2} \) |
| 37 | \( 1 - 5.83T + 37T^{2} \) |
| 41 | \( 1 + 0.306T + 41T^{2} \) |
| 43 | \( 1 - 0.173T + 43T^{2} \) |
| 47 | \( 1 - 3.13T + 47T^{2} \) |
| 53 | \( 1 + 1.76T + 53T^{2} \) |
| 59 | \( 1 + 3.38T + 59T^{2} \) |
| 61 | \( 1 - 5.00T + 61T^{2} \) |
| 67 | \( 1 - 3.68T + 67T^{2} \) |
| 71 | \( 1 + 14.8T + 71T^{2} \) |
| 73 | \( 1 + 1.26T + 73T^{2} \) |
| 79 | \( 1 + 0.660T + 79T^{2} \) |
| 83 | \( 1 + 6.79T + 83T^{2} \) |
| 89 | \( 1 - 3.21T + 89T^{2} \) |
| 97 | \( 1 + 7.76T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.911387986013326176531030965441, −7.04263017380571987413472878904, −6.18543397124875100474630642095, −5.92000162740192757053659304891, −5.19069981904271486684403411570, −4.41663283944970326018698613727, −3.98083337884918685448104286786, −2.79676378246985703421488614539, −2.08381284125319149521502082941, −1.52837932348861430373871383312,
1.52837932348861430373871383312, 2.08381284125319149521502082941, 2.79676378246985703421488614539, 3.98083337884918685448104286786, 4.41663283944970326018698613727, 5.19069981904271486684403411570, 5.92000162740192757053659304891, 6.18543397124875100474630642095, 7.04263017380571987413472878904, 7.911387986013326176531030965441
